Motivated by some conjectures originating in the physics literature, we use Foscolo's construction of Ricci-flat K & auml;hler metrics on K3 surfaces to locate, with high precision, several closed geodesics and compute their index (their length is also approximately known). Interestingly, the construction of these geodesics is related to an open problem in electrostatics posed by Maxwell in 1873. Our construction is also of interest to modern physicists working on (supersymmetric) non-linear sigma models with target space such a K3 surface.
Abstract Let F 0 F_{0} be a complete discretely valued field with residue field a global field with no real places or a local field. Let F / F 0 F/F_{0} be a separable quadratic extension and 𝐴 a central simple algebra over 𝐹 with an F / F 0 F/F_{0} -involution 𝜏. Let ℎ be a hermitian form over ( A , τ ) (A,\tau) . If 2 ind ( A ) 2\operatorname{ind}(A) is coprime to the characteristic of the residue field of F 0 F_{0} , then we prove that the Rost invariant R SU ( h ) : H 1 ( F 0 , SU ( h ) ) → H 3 ( F 0 , Q / Z ( 2 ) ) R_{\operatorname{SU}(h)}\colon H^{1}(F_{0},\operatorname{SU}(h))\to H^{3}(F_{0},\mathbb{Q}/\mathbb{Z}(2)) is injective. Let 𝐾 be a local field and 𝐿 the function field of a curve over 𝐾. Let 𝐺 be an absolutely simple simply connected group over 𝐿 of classical type. Suppose that the characteristic of the residue field of 𝐾 is a good prime for 𝐺. As a consequence of our result and some known results, we conclude that the Rost invariant of 𝐺 is injective.
Abstract For a non-Archimedean locally compact field 𝐹 of odd residue characteristic and characteristic 0, we prove a conjecture of Prasad predicting that, for an integer n ⩾ 1 n\geqslant 1 and a non-split quaternionic 𝐹-algebra 𝐷, a discrete series representation of GL n ( D ) \operatorname{GL}_{n}(D) has a symplectic period if and only if it is cuspidal and its Jacquet–Langlands transfer to GL 2 n ( F ) \operatorname{GL}_{2n}(F) is non-cuspidal.
We introduce a sufficient condition for a finitely generated subgroup Gamma of a semisimple Lie group G to admit finite-sided Dirichlet domains for polyhedral Finsler metrics on the symmetric space G / K G/K . The condition always implies the Theta-Anosov condition for some Theta, and can be arranged to be equivalent to the Theta-Anosov condition when G is simple and Theta is the set of long roots or the set of short roots. The Dirichlet domain we obtain extends to a fundamental domain for the action of Gamma on a domain of discontinuity in a flag manifold. For instance, Borel-Anosov subgroups of SL (d , R ) have finite-sided Dirichlet domains for the Hilbert metric on the symmetric space which extends to the space of line-hyperplane flags, and n-Anosov subgroups of Sp(2n, R) have finite-sided Dirichlet-Selberg domains in SL(2n, R)/SO(2n) which extend to a domain in projective space bounded by quadrics.
We investigate the Picard group of the Baily-Borel compactification of orthogonal Shimura varieties. In particular, we show that the Picard group of the Baily-Borel compactification of the moduli space of quasi-polarized K3 surfaces is isomorphic to & Zopf;. Notably, this contrasts with the moduli space of smooth curves, whose natural compactification has a more complicated Picard group. More generally, we show that, for an even lattice M of signature (2,n) with n > 8 satisfying certain splitting conditions, the rational Picard group of Sh(Gamma)( M ) has dimension 1 when Gamma contains the stable orthogonal group. The obstruction space that governs the extension of Heegner divisors on orthogonal Shimura varieties to the boundary can be identified with the space spanned by certain theta series. The central argument of our proof is to show that this space is maximal under our conditions.
